Question

f（x）是连续函数,满足f(x)=exp{∫f(t/3)dt},积分上限是3x ,下限是0,求f(x

Answer 1

f(x)=exp{∫(0,3x)f(t/3)dt}

两边同时求导得

f'(x)=exp{∫(0,3x)f(t/3)dt}* (∫(0,3x)f(t/3)dt)'

=f(x)*f(x)*3

=3f²(x)

所以

【1/f²(x)】df(x)=3dx

-1/f(x)=3x+c

f(x)=-1/(3x+c)

又x=0时,f(0)=1

所以

1=-1/c

c=-1

即

f(x)=-1/(3x-1)